There is also a close relationship between potential theory in the plane and complex analysis: concepts from potential theory are important and natural tools for the study of holomorphic functions. Further, this connection suggests potential theoretic analogues of theorems concerning functions of one complex variable, ranging from elementary results such as the maximum modulus theorem and Laurent's theorem, to the approximation theorems of Runge and Mergelyan and the theory of prime ends. We treat our subject at a level intended to be accessible to graduate students. Prerequisite knowledge does not go beyond what is commonly taught in undergraduate or first-year graduate courses. The reader will need a good grasp of the limiting processes of analysis, some facility with calculus in higher dimensions, and some measure theory.